Step 1: Check Statement I.
The carbonates of group 2 elements are generally insoluble in water.
Hence,
\[
\boxed{\text{Statement I is correct}.}
\]
Step 2: Check Statement II.
The solubility of sulphates decreases down the group:
\[
\boxed{\mathrm{CaSO_4}>\mathrm{SrSO_4}>\mathrm{BaSO_4}.}
\]
Hence,
Statement II is correct.
Step 3: Check Statement III.
On heating,
\[
\mathrm{M(NO_3)_2}
\rightarrow
\mathrm{MO}+2\mathrm{NO_2}+\frac12\mathrm{O_2}.
\]
Thus, group 2 nitrates undergo thermal decomposition.
Hence,
Statement III is also correct.
Therefore,
\[
\boxed{\text{Statements I, II and III are correct}.}
\]
Hence, the correct option is \(\boxed{(D)}\).